# let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f$ uniformly on $[0, \infty)$

this sequence was given as a practice problem and I'm really having trouble. Heres the question:

let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f$ uniformly on $[0, \infty)$

My Work: I determined that the pointwise limit $f(x) =$lim$f_n(x)$ has to be $1$ but I am not too sure what to do after this. Any help is appreciated

Ans No, it is not. It is $0$ in $[0,1)$, $1/2$ at $1$ and $1$ over $(1,\infty]$.
you can use the following characterization : $f_n$ converges uniformly to $f$ on $S$ iff
$$\lim_{n \to \infty} \sup_{x \in S} \{ |f_n(x) - f(x) | \} = 0$$